arXiv subjects

Jimmie Adriazola

Publications and source records attributed to Jimmie Adriazola.

10 recordsLinked to original sources

arXiv2026

On the dispersionless limit of the Manakov system, its Riemann invariants, and the modulational stability of its counterpropagating plane waves

We study the dispersionless limit of the Manakov system, the integrable two-component generalization of the nonlinear Schrödinger equation. We derive the resulting four-component genus-zero Manakov-Whitham system, characterize its hydrodynamic structure, and show that it passes the Haantjes tensor test for integrability. We show that the branch points of the spectral curve associated with plane wave solutions of the Manakov system are the local Riemann invariants of the dispersionless system. We also use the characteristic speeds to classify the baseband modulational stability/instability of the plane waves, and we study a direct linearization of the Manakov system to characterize their finite-wavenumber stability and verify agreement with the Whitham prediction in the long-wave limit. Finally, we validate the predictions by comparing them with the results of direct numerical simulations.

nlin.SI↗

arXiv2026

Machine Learning of Nonlinear Waves: Data-Driven Methods for Computer-Assisted Discovery of Equations, Symmetries, Conservation Laws, and Integrability

The purpose of this article is to provide a perspective---admittedly, a rather subjective one---of recent developments at the interface of machine learning (ML)/data-driven methods and nonlinear wave studies. We review some recent pillars of the rapidly evolving landscape of scientific ML, including deep learning, data-driven equation discovery, Koopman-based methods, and operator learning, among others. We then showcase these methods in applications ranging from learning lattice dynamical models and reduced order modeling of effective dynamics to discovery of conservation laws and potential identification of integrability of ordinary differential equations (DEs) and partial DE models. Our intention is to make clear that these ML methods are complementary to the preexisting powerful tools of the nonlinear waves community, and should be integrated into this toolkit to augment and enable mathematical discoveries and computational capabilities in the age of data.

nlin.PS↗

arXiv2026

Learning Volterra Memory Kernels for Non-Markovian Qubit Dynamics

We develop a data-driven framework for identifying non-Markovian equations of motion for open quantum systems, demonstrated here for qubit-environment dynamics. Starting from the Nakajima-Zwanzig formalism, we vectorize the reduced density matrix into a four-dimensional state vector and cast the dynamics as a Volterra integro-differential equation with an operator-valued memory kernel. The learning task is then formulated as a constrained optimization problem over the admissible operator space, where correlation functions are approximated by rational functions using Pade approximants. We establish well-posedness of the learning problem, ensuring existence of minimizers. To assess performance, we construct synthetic data sets from representative test problems of increasing complexity: (i) exactly solvable pure dephasing, with correlation functions expressed in terms of special functions, (ii) a damped Jaynes-Cummings model with an analytic coherence kernel, (iii) a transverse Born model with frequency-resolved bath integrals and population-coherence coupling, and (iv) a non-rotating-wave quantum Rabi model whose memory kernel has no closed form. Numerical experiments demonstrate that Pade captures nontrivial temporal structures such as oscillatory memory, algebraic tails, and phase-sensitive coherence transfer, and that the learned models generalize across ensembles of physically admissible initial states. We perform a parametrization-invariant sensitivity analysis and show that the trajectories are insensitive to the unrecoverable parts of the kernel, so the learned models stay predictive despite severe ill-conditioning in kernel recovery. These results together illustrate that data-driven rational approximation provides an effective route to identifying non-Markovian kernels of practical relevance in quantum technologies.

quant-ph↗

arXiv2026

Learning Lax Pairs: Revisiting the Classical Paradigm

A Lax pair $(L,P)$ is sometimes thought of as a structural certificate, in that the spatial operator $L$ carries the spectral data of an integrable system, and its isospectral evolution under $\partial_t L = [L,P]$ encodes the nonlinear dynamics. Yet, experience shows that the correspondence between equations and Lax pairs is much more nuanced than this picture suggests. Equations can admit Lax pairs that fail to encode the expected integrable structure. This paper probes that anomalous corner of the Lax pair landscape through five case studies (the Euler top, the free Schrödinger equation, the inviscid Burgers equation, the shallow water system, and the Korteweg--de Vries equation), each illustrating a different way the link to integrability can be distorted. The approach combines analytical calculations with the Sparse Identification of Lax Operators (SILO) framework, which proved useful throughout, in some cases confirming the textbook pair and in others surfacing alternatives worth understanding on their own terms. The recurring lesson across the five cases is that compatibility underdetermines the Lax representation, so that anomalous pairs are regular features of the landscape rather than pathologies. Notably, we show that a spectrally degenerate Korteweg--de Vries Lax pair, classified as fake by standard criteria, still generates the full conservation hierarchy through its operator algebra, which shows that a blunt dichotomy between true and fake Lax pairs can be too reductive.

nlin.SI↗

arXiv2026

Structure-Preserving Optimal Control of Maxwell's Equations with Applications to Source Cloaking

We develop a structure-preserving solution framework for the optimal control of the time-dependent Maxwell's equations. Building on a well-posedness theory for a weak form of the forward problem, we first analyze a forward solver that couples Nédélec and Raviart--Thomas finite elements with Crank--Nicolson time stepping. The solver preserves the de~Rham structure, enforces a discrete Gauss law, exactly satisfies a per-time-step energy balance, and converges to the weak solution under low regularity assumptions on the problem data, which are dictated by the optimal control setting. To control the Maxwell system, we add the curl of a space-time current density as a source to Ampére's law. The curl form yields charge conservation without auxiliary constraints. We prove the well-posedness and continuity of the control-to-state map, derive the adjoint system and a gradient representation for a tracking-type objective functional, and formulate a discrete optimization scheme that inherits structure preservation from the forward solver. Our discrete stationarity conditions are consistent with their continuous counterparts, and the discrete optimal controls converge, with mesh and time refinements, to the continuous optima. We demonstrate the merits of our optimal control formulation and the theoretical developments by numerically solving a series of source-cloaking model problems.

math.OC↗

arXiv2025

Computer Assisted Discovery of Integrability via SILO: Sparse Identification of Lax Operators

We formulate the discovery of Lax integrability of Hamiltonian dynamical systems as a symbolic regression problem, which, loosely speaking, seeks to maximize the compatibility between a pair of Lax operators and the known Hamiltonian of the dynamical system. Our approach is first tested on the simple harmonic oscillator. We then move on to the Henon-Heiles system, i.e. a two-degree-of-freedom system of nonlinear oscillators. The integrability of the Henon-Heiles system is critically dependent on a set of three parameters within its Hamiltonian, a fact that we leverage to assess the robustness of our approach in detecting the integrability of this system with respect to the parameter dependence of the Hamiltonian. We then adapt our method to canonical examples of Hamiltonian partial differential equations, including the Korteweg-de Vries and cubic nonlinear Schrödinger equations, again testing robustness against nonintegrable perturbations of their respective Hamiltonians. In all examples, our approach reliably confirms or denies the integrability of the equations of interest. Moreover, by appropriately adjusting the loss function and applying thresholded $l^0$ regularization to enforce sparsity in the operator weights, we successfully recover accurate forms of the Lax pairs despite wide initial hypotheses on the operators. Some of the relevant Lax pairs, notably for the Henon-Heiles system and the Korteweg-deVries equation, are distinct from the ones that are typically reported in the literature. The Lax pairs that our methodology discovers warrant further mathematical and computational investigation, and we discuss extensively the opportunities for further improvement of SILO as a viable tool for interpretable exploration of integrable Hamiltonian dynamical systems.

nlin.SI↗

arXiv2025

Experimentally Tractable Generation of High-Order Rogue Waves in Bose-Einstein Condensates

In this work, we study a prototypical, experimentally accessible scenario that enables the systematic generation of so-called high-order rogue waves in atomic Bose-Einstein condensates. These waveforms lead to significantly and controllably more extreme focusing events than the famous Peregrine soliton. In one spatial dimension, we showcase conclusive numerical evidence that our scheme generates the focusing behavior associated with the first four rogue waves from the relevant hierarchy. We then extend considerations to anisotropic two-dimensional and even three-dimensional settings, establishing that the scheme can generate second order rogue waves despite the well-known limitation of finite-time blow up of focusing nonlinear Schrödinger equations.

nlin.PS↗

arXiv2024

A Non-Convex Optimization Strategy for Computing Convex-Roof Entanglement

We develop a numerical methodology for the computation of entanglement measures for mixed quantum states. Using the well-known Schrödinger-HJW theorem, the computation of convex roof entanglement measures is reframed as a search for unitary matrices; a nonconvex optimization problem. To address this non-convexity, we modify a genetic algorithm, known in the literature as differential evolution, constraining the search space to unitary matrices by using a QR factorization. We then refine results using a quasi-Newton method. We benchmark our method on simple test problems and, as an application, compute entanglement between a system and its environment over time for pure dephasing evolutions. We also study the temperature dependence of Gibbs state entanglement for a class of block-diagonal Hamiltonians to provide a complementary test scenario with a set of entangled states that are qualitatively different. We find that the method works well enough to reliably reproduce entanglement curves, even for comparatively large systems. To our knowledge, the modified genetic algorithm represents the first derivative-free and non-convex computational method that broadly applies to the computation of convex roof entanglement measures.

quant-ph↗

arXiv2021

An Optimal Control Approach to Gradient-Index Design for Beam Reshaping

We address the problem of reshaping light in the Schrödinger optics regime from the perspective of optimal control theory. In technological applications, Schrödinger optics is often used to model a slowly-varying amplitude of a para-axially propagating electric field where the square of the waveguide's index of refraction is treated as the potential. The objective of the optimal control problem is to find the controlling potential which, together with the constraining Schrödinger dynamics, optimally reshape the intensity distribution of Schrödinger eigenfunctions from one end of the waveguide to the other. This work considers reshaping problems found in work due to Kunkel and Leger, and addresses computational needs by adopting tools from the quantum control literature. The success of the optimal control approach is demonstrated numerically.

math.OC↗

arXiv2021

A Reduction-Based Strategy for Optimal Control of Bose-Einstein Condensates

Applications of Bose-Einstein Condensates (BEC) often require that the condensate be prepared in a specific complex state. Optimal control is a reliable framework to prepare such a state while avoiding undesirable excitations, and, when applied to the time-dependent Gross-Pitaevskii Equation (GPE) model of BEC in multiple space dimensions, results in a large computational problem. We propose a control method based on first reducing the problem, using a Galerkin expansion, from a PDE to a low-dimensional Hamiltonian ODE system. We then apply a two-stage hybrid control strategy. At the first stage, we approximate the control using a second Galerkin-like method known as CRAB to derive a finite-dimensional nonlinear programming problem, which we solve with a differential evolution (DE) algorithm. This search method then yields a candidate local minimum which we further refine using a variant of gradient descent. This hybrid strategy allows us to greatly reduce excitations both in the reduced model and the full GPE system.

cond-mat.quant-gas↗